Research in Differential Geometry and Topology

本文将研究R3中适当嵌入的最小曲面的几何、渐近行为、共形结构和拓扑结构。该建议的主要目标之一是对所有适当嵌入的零格最小曲面进行分类，并描述所有有限格实例的渐近几何。研究了局部有界属最小曲面的紧性、正则性和收敛性的相关理论技术。本研究的一个应用希望是对S3上所有光滑有限群作用进行分类。最后，研究提出证明双曲三维中的Bryant曲面是解结的。经典的最小曲面理论起源于18世纪和19世纪的数学，并在所谓的变分学中给出了最早的重要例子之一，这种变分学最早由欧拉描述。物理上最小的表面可以局部建模为电线上的肥皂膜或相对于局部边界的最小面积表面。这些曲面作为研究三维拓扑和黎曼几何的工具发挥着重要作用。本提案的研究涉及这些表面的整体特性以及在三维拓扑和几何的基础研究中的可能应用。

Abstract for DMS - 0104044In this proposal the research will study the geometry, asymptotic behavior, conformal structure and topology of properly embedded minimal surfaces in R3 . One of the main goals of the proposal is to classify all properly embedded minimal surfaces of genus zero and to describe the asymptotic geometry of all finite genus examples. Related theoretical techniques concerning compactness, regularity and convergence of minimal surfaces of locally bounded genus will be investigated as well. One hoped for application of this research is to classify all smooth finite group actions on S3 . Finally the research proposes to prove that Bryant surfaces in hyperbolic three-space are unknotted.Classical minimal surface theory has its roots in 18th and 19th century mathematicsand gives one of the first important examples in what is called the calculus of variations, first described by Euler. Physically minimal surfaces can be modeled locally as soap films on wires or by surfaces of least-area relative to local boundaries. These surfaces play an important role as a tool in the study ofthree-dimensional topology and Riemannian geometry. The research in this proposal concerns global properties of these surfaces and possible applications to basic research in three-dimensional topology and geometry.